RESISTIVITY MINIMUM IN AMORPHOUS FERROMAGNETS

HAL Id: jpa-00215645

https://hal.archives-ouvertes.fr/jpa-00215645

Submitted on 1 Jan 1974

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

RESISTIVITY MINIMUM IN AMORPHOUS FERROMAGNETS

A. Madhukar, R. Hasegawa

To cite this version:

A. Madhukar, R. Hasegawa. RESISTIVITY MINIMUM IN AMORPHOUS FERROMAGNETS.

Journal de Physique Colloques, 1974, 35 (C4), pp.C4-291-C4-293. �10.1051/jphyscol:1974454�. �jpa- 00215645�

JOURNAL DE PHYSIQUE Colloque C4, suppliment au no 5, Tome 35, Mai 1974, page C4-291

RESISTIVITY MINIMUM IN AMORPHOUS FERROUGNET S

A. MADHUKAR and R. HASEGAWA IBM Thomas J. Watson Research Centre Yorktown Heights, New York 10598, U. S. A.

RBsumB. - Nous proposons un mkcanisme de collisions avec renversement de spin entre electrons dans les solides ferromagnktiques amorphes et montrons que ce mecanisme permet de rendre compte du minimum de resistivite observe dans ces solides.

Abstract. ^- The puzzling phenomenon of resistivity minimum observed below the magnetic ordering temperature in amorphous ferromagnets is explained on the basis of electron spin flip scattering off localised magnon states. Such localised magnon states exist at very low energies and are characteristic of the amorphous state.

In the recent past investigations of certain amor- phous ferromagnets have revealed the existence of a resistivity minimum well below the magnetic order- ing temperature 11-51. The well accepted explanation for a resistivity minimum in very dilute magnetic alloys (amorphous as well as crystalline) is based upon the dynamical nature of the isolated magnetic impu- rity [6]. However, such an explanation does not hold for the highly concentrated magnetic alloys under consideration here. Well below the magnetic ordering temperature the local spins are expected to be ordered and as such spin flip scattering of conduction electrons by these spins is expected to be quenched. This beha- viour is characteristic of crystalline alloys, where increase of magnetic atom concentration leads to the disappearance of the resistivity minimum. Thus a new mechanism inherent to the amorphous state of these highly concentrated magnetic alloys, has to be sought to explain the resistivity minimum.

It is the purpose of this paper to show that a mechanism for conduction electron spin flip scatter- ing is provided by localised magnon states of energy ko < k, T, characteristic of the amorphous state.

This spin flip scattering leads to a temperature depen- dent contribution to the resistivity, which when combined with the normal part, gives rise to a resis- tivity minimum.

Although no rigorous calculation of the magnon density of states for amorphous magnets exists in the literature, in figure 1, we have schematically shown the postulated and expected behaviour. This is support- ed by a perturbation theory calculation of the magnon density of states for disordered systems 171. The peak at low energies and the tail at high energies is found to be characteristic of disordered systems. With

FIG. 1. - Schematic diagram of the magnon dispersion curve for crystalline (solid line) and amorphous (cross hatched area)

states.

increasing disorder, it was further found that the low energy peak shifts to lower and lower energies. Thus we postulate the existence of this low energy peak, at energies less than the temperature (T -- 25 K) at which the resistivity minimum is observed. These localised magnon states provide the angular momen- tum necessary for conduction electron spin flip scattering, thereby giving rise to the observed resis- tivity minimum.

We note also that many of these amorphous alloys exhibit a clustering of magnetic atoms. As such, we would expect the localised spin deviations to have higher momentum components and thus a higher energy. However, the reduced spin stiffness coefficient

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974454

C4-292 A. MADHUKAR AND R. HASEGAWA

characteristic of the amorphous state allows a mixing (- hop) involved in (2) in evaluating the sum. This of higher momentum components without a significant restricts the validity of our calculations to

raising of the energy of these localised states. This

is schematically shown in figure 2. k, T > hw,.

We shall, a posteriori, show that this neglect is justified in the temperature region of the resistivity minimum.

We find that,

where z, is the transport relaxation time and S is the structure factor defined as,

FIG. 2. - Schematic diagram for the density of states for we evaluate this by writing,

crystalline (solid) and amorphous (dashed) phases. Note the

peak at low energies for the amorphous phase. G = S(2 k - k)

+

S"(2 k" - k) The exchange interaction of the local spin and con- ₌

5 Ia

R' cos (2 k" - kR) g(R) dR duction electrons is written as a contact interaction, ^{9 0} o

where ^o and S are the conduction electron spin density

and the local spin value at Rj. The suffix j on J~ allows Here

no

is the volume over which integration is per- for the possibility of spatial of the exchange formed. The resistivity is calculated using the standard interaction strength either due to disorder or due to Boltzman's expression, with the following result the presence of more than one kind of magnetic spin,

or both. However, for the purpose of illustrating the Ap = [ a

+

^{yG In}

(y) ^{] .z}

^(n,

⁺

¹⁾

^.

⁽⁶⁾

origin of the resistivity minimum it suffices to consider ⁴

a single value of J and S. The sum over j is understood to be an average over space, performed using the radial distribution function as the weighting factor.

This averaging is done at the end of all calculations.

We calculate the conduction electron transition probability, W(ko -+ k'o'), to second order in Born approximation, retaining terms involving the Fermi factors only (analogous to Kondo). In doing so, we encounter the usual sum over intermediate states,

where n, and f,,, are the magnon and electron occu- pation functions. As discussed before, the sum over q is heavily weighted on the low energy side due to the replacement

Thus we neglect the characteristic magnon energy

In (6), a and y are the usual Kondo coefficients given in terms of J, S, cF and the number of conduction electrons per atom. The logarithmic temperature dependent term, when combined with the normal part of the resistivity gives rise to a resistivity minimum

for J < 0. However, we notice that the usual Kondo

behaviour of the resistivity is modified in two respects ; (i) the structure factor G reduces the Kondo coefficient, y ; (ii) the usual Kondo type temperature dependence is further modified by the temperature dependence arising from the Bose statistics of the local spin devia- tions as reflected in,

To evaluate (7) we need to know the explicit form of the magnon density of states, since this affects the temperature dependence of (7). Since no analytic expression for p,,,(o,) exists, we chose a triangle

RESISTIVITY MINIMUM IN AMORPHOUS FERROMAGNETS C4-293

density of states which approximates the behaviour and, h o p

-

^{5 K ,} we find that (kB T/hom) < 10-I shown in figure 2 exceedingly well. Thus we choose, and ( h o p / k , T )

-

115. Thus in the region of the resis-

tivity minimum substituting from (9) into (6) we find ( p p i o p ) ; 0 G o

<

^op that the resistivity behaves like,

( ~ ~- i~ p ) w^{( o m} ~- ~0 ) ; ~ (8)

u p G a

<

amax ^Ap

-

^{( A}

⁺

B l n T ) . (10) where o, is the energy a t which pm,,(o) takes its peak

value and omax is the maximum magnon energy, which we take to be of the order of the magnetic ordering

temperature. Substituting (8) into (7) we evaluate (7)

-

^0.8

under the assumption (hwp/homax)

<

1 and

C

with the following result ;

r

and 5 are the Gamma and Reimann Zeta functions TEMPERATURE ( K)

respectively. Since typically the magnetic ordering _{FIG. 3. -} Resistivity Ap as a function of temperature. The temperature for these materials is

-

^{300 K or higher} dashed line for 3 c T < 30 K corresponds to the expression the resistivity minimum is observed around T

-

^{25 K Ap = A f} Bln T where A = 1.14 and B = - 349 D-cm.

References

[I] LIN, S. C. H., J. Appl. Phys. 40 (1969) 2173. 151 HASEGAWA, R., J. Appl. Phys. 41 (1970) 4096.

[2] MAITREPIERRE, P. L., Ph. D. Thesis, California Institute of [6] KONDO, J., in Solid State Physics, ed. F. Sertz and D. Turn- Technology, Pasadena, California 91109 (1969). bull (Academic Press, N. Y.) 1969, vol. 23, chap. 2, [3] HASEGAWA, R. and Tsu~r, C. C., Phys. Rev. B 3 (1971) 214. and HEEGER, A. J., ibid., chap. 3.

[4] HASEGAWA, R. and DERMON, J. A., Phys. Lett. 42A (1973) [7] MONTGOMERY, C. G., KRUGLER, J. I. and STUBBS, R. M.,

407 Phys. Rev. Lett. 25 (1970) 669.